In this paper the following is proved: Let $\mathcal{L}$ be a subspace
lattice on a Hilbert space $\mathcal{H}$ and X and Y be operators acting on
$\mathcal{H}$. Then
there exists a compact operator A in Alg$\mathcal{L}$ such that AX = Y if and only
if
$\sup \left\{ {\norm{E^\perp Y f} \over \norm{E^\perp X f}} : f \in
\mathcal{H}, ~E \in \mathcal{L}
\right\} = K < \infty$ and $Y$ is compact. Moreover,
if the necessary condition holds, then we may choose an operator $A$ such
that $AX = Y$ and $\norm{A} = K$.